Extended chiral Khuri–Treiman formalism for $$\eta \rightarrow 3\pi $$ η → 3 π and the role of the $$a_0(980)$$ a 0 ( 980 ) , $$f_0(980)$$ f 0 ( 980 ) resonances

Abstract Recent experiments on $$\eta \rightarrow 3\pi $$ η → 3 π decays have provided an extremely precise knowledge of the amplitudes across the Dalitz region which represent stringent constraints on theoretical descriptions. We reconsider an approach in which the low-energy chiral expansion is assumed to be optimally convergent in an unphysical region surrounding the Adler zero, and the amplitude in the physical region is uniquely deduced by an analyticity-based extrapolation using the Khuri–Treiman dispersive formalism. We present an extension of the usual formalism which implements the leading inelastic effects from the $$K\bar{K}$$ K K ¯ channel in the final-state $$\pi \pi $$ π π interaction as well as in the initial-state $$\eta \pi $$ η π interaction. The constructed amplitude has an enlarged region of validity and accounts in a realistic way for the influence of the two light scalar resonances $$f_0(980)$$ f 0 ( 980 ) and $$a_0(980)$$ a 0 ( 980 ) in the dispersive integrals. It is shown that the effect of these resonances in the low-energy region of the $$\eta \rightarrow 3\pi $$ η → 3 π decay is not negligible, in particular for the $$3\pi ^0$$ 3 π 0 mode, and improves the description of the energy variation across the Dalitz plot. Some remarks are made on the scale dependence and the value of the double quark mass ratio Q.